Marco Belinelli hit a 3-pointer with less than a minute left to cut Chicago's deficit to 89-87, but Paul answered with a floater in the lane, then iced it with three free throws down the stretch as Los Angeles snapped the Bulls' season-high three-game winning streak.
Paul finished with 18 points and four assists, less than the 20 and 10 he had averaged in his first dozen games against Chicago, but enough for the Clippers to sweep the season series for the first time since 2009-10.
"I kind of forgot what it was like to play in the fourth quarter," said Paul, who hadn't seen late action since Dec. 3 at Utah. "But at the end, we knew we were on the road and we'd have to withstand a run we knew they would make. Our defense kept us in the game."
Chicago shot 50 percent from beyond the arc but only 1-of-6 in the fourth quarter.
"We want our identity to be a defensive identity," Paul said. "We're still trying to build that."
The Clippers (15-6) are on their longest winning streak since an eight-game run during the 1991-92 season. They have won 10 of 12 at the United Center.
Carlos Boozer scored 24 points and pulled down 13 rebounds for the Bulls, who had won five of six overall.
It was Boozer's 10th double-double in 13 games, but he missed a pair of free throws and was called for an offensive foul during a crucial sequence three minutes into the fourth quarter as Chicago failed to cut into the Clippers' lead.
The Bulls (11-9) had held opponents to fewer than 90 points in seven of their previous nine games, although they did keep the Clippers to fewer than 100 for the first time during their seven-game run.
Griffin was a force inside throughout, quieting the home crowd with his thunderous dunks. He also knocked down his first 3-pointer of the season in the second quarter after missing five previous attempts.
"Games on the road like this, after we had a little bit of success, six games, coming on the road, you don't want to have a letdown," Griffin said. "Chicago's a tough team. These are the kinds of wins you need."
The Western Conference player of the week followed up his 3 with a pair of rim-shaking dunks in transition as part of a 15-5 Clippers run to end the first half. They went into halftime with a seven-point lead, matching their biggest advantage of the game to that point.
"The end of the second quarter changed the game," Bulls coach Tom Thibodeau said. "We started dancing with the ball. When you do that, it leads to turnovers, live ball and transition baskets for them. We shot ourselves in the foot there.
"You can't do that against good teams."
Griffin averaged 30.3 points and 12.0 rebounds in his first four career games against Chicago. The only team he has better numbers against is the New York Knicks (31.1 ppg).
Griffin had 26 points and 10 rebounds in his team's 101-80 win over Chicago on Nov. 17 in Los Angeles.
Joakim Noah finished with 10 points and 11 rebounds. It marked the first time the Bulls have lost when both Noah and Boozer have had double-doubles (8-1).
Luol Deng was held to just eight points, his lowest output since a seven-point performance in Chicago's season opener.
He and Noah committed four turnovers apiece. Chicago turned the ball over 16 times.
"We can't put that team in transition," Noah said. "The turnovers hurt us."
Defense ruled early as both teams struggled out of the chute. Los Angeles missed its first six field goal attempts before Caron Butler finally knocked down a jumper more than 3 minutes in. Chicago fared little better, starting 1-of-10 from the floor.
Robinson knocked down a pair of 3s to help Chicago keep pace, and Gibson's jumper midway through the second quarter gave the Bulls their first lead since the opening minutes.
Gibson scored eight of his 10 points in the second quarter.
Jamal Crawford, the NBA's leading bench scorer, finished with 10 points for Los Angeles. They all came in the second after he was tripped by Robinson, who was called for a flagrant foul.
Clippers coach Vinny Del Negro said it was "a little bit" surprising that Griffin's player of the week award, announced Monday, was his first such honor since entering the league in 2010. "Those things happen when you win," Del Negro said. "You get recognized for winning, and I think Blake is most pleased with that." ... Clippers F Grant Hill (right knee bone bruise) "felt pretty good" after running Monday, according to Del Negro. Hill will continue to run every other day, but there still is no timetable for his return. ... Hinrich (left elbow) was able to start after missing the second half of Saturday's win against New York. "It says a lot about his toughness," Thibodeau said. "I don't measure him statistically. I measure by how he's running the team, how he's leading our defense."